So we’ve introduced, with maybe more words than strictly necessary, the idea that we can set up a match between the numbers from 0 to 496 and particular locations on the New York Thruway. There are a number of practical quibbles that can be brought against this scheme. For example: could we say for certain that the “outer” edge of this road, which has roughly the shape of an upside-down u, isn’t loger than the “inner” edge? We may need more numbers for the one side than the other. And the mile markers, which seemed like an acceptable scheme for noting where one was, are almost certainly only approximately located.
But these aren’t very important. We can imagine the existence of the “ideal” Thruway, some line which runs along the median of the whole extent of the highway, so there’s no difference in length running either direction, and we can imagine measuring it to arbitrarily great precision. The actual road approximates that idealized road. And this gives what I had really wanted, a kind of number line. All the numbers from zero to 496 (or so) match a point on this ideal Thruway line, and all the points on this Thruway match some number between zero and 496. That the line wriggles all over the place and changes direction over and over, well, do we really insist that a line has to be straight?
Well, we can at least imagine taking this “ideal” Thruway, lifting it off the globe and straightening it out, if we really want to. Here we invoke a host of assumptions even past the idea that we can move this curvy idealized road around. We assume that we can straighten it out without changing its length, for example. This isn’t too unreasonable if we imagine this curve as being something like a tangled bit of string and that we straighten it out without putting any particular tension on it; but if we imagined the idealized road as being a rubber band, held taut at the New York City and Ripley, New York, ends and pinned in place at the major turns we notice that isn’t actually guaranteed. Let’s assume we can do this straightening-out without distorting the lengths, though.